sinx+sin2x+sin3x=sin(2x-x)+sin2x+sin(2x+x)
=sin2xcosx-cos2xsinx+sin2x+sin2xcosx+cos2xsinx
=2sin2xcosx+sin2x
=(2cosx+1)sin2x
=0
所以cosx=-1/2或sin2x=0,
若sin2x=0,则2sinxcosx=0,
即sinx=0(因为tanx=a存在,所以cosx≠0),
cosx=1,tanx=0；
若cosX=-1/2,则sinX=±√3/2,
所以tanX=±√3
综上,a=0或±√3.